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href="#4-考题：2011-7、2012-4、2013-3-PDF中的分析有误，以上课讲解为准-、2013-6、2015-4、2018-6、2019-4、2019-6、2020-5、2017-41"><span class="toc-text">4.考题：2011-7、2012-4、2013-3(PDF中的分析有误，以上课讲解为准)、2013-6、2015-4、2018-6、2019-4、2019-6、2020-5、2017-41</span></a></li></ol></li></ol></div></div></div></aside><main class="sidebar-translate" id="content"><div id="post"><article class="post-block" itemscope itemtype="https://schema.org/Article"><link itemprop="mainEntityOfPage" href="http://yoursite.com/2021/07/19/%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84/acwing/sjjg_4_2/"><span hidden itemprop="author" itemscope itemtype="https://schema.org/Person"><meta itemprop="name" content="李子康"><meta itemprop="description"></span><span hidden itemprop="publisher" itemscope itemtype="https://schema.org/Organization"><meta itemprop="name" content="Lizikang_Blog"></span><header class="post-header"><h1 class="post-title" itemprop="name headline">数据结构 第5讲 二叉排序树、表达式树</h1><div class="post-meta"><div class="post-time" style="display:inline-block"><span class="post-meta-item-icon"><svg class="icon" aria-hidden="true"><use xlink:href="#icon-calendar-line"></use></svg></span> <time title="创建时间：2021-07-19 00:00:00" itemprop="dateCreated datePublished" datetime="2021-07-19T00:00:00+08:00">2021-07-19</time><span class="post-meta-divider">-</span><span class="post-meta-item-icon"><svg class="icon" aria-hidden="true"><use xlink:href="#icon-calendar-2-line"></use></svg></span> <time title="修改时间：2021-08-25 14:29:48" itemprop="dateModified" datetime="2021-08-25T14:29:48+08:00">2021-08-25</time></div><span class="post-busuanzi"><span class="post-meta-divider">-</span><span class="post-meta-item-icon" title="阅读次数"><svg class="icon" aria-hidden="true"><use xlink:href="#icon-eye-line"></use></svg> <span id="busuanzi_value_page_pv"></span></span></span><div class="post-classify"><span class="post-category"><span class="post-meta-item-icon"><svg class="icon" aria-hidden="true"><use xlink:href="#icon-folder-line"></use></svg></span> <span itemprop="about" itemscope itemtype="https://schema.org/Thing"><a class="category" href="/categories/%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84/" style="--text-color:var(--hty-text-color)" itemprop="url" rel="index"><span itemprop="text">数据结构</span></a></span> > <span itemprop="about" itemscope itemtype="https://schema.org/Thing"><a class="category" href="/categories/%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84/acwing/" style="--text-color:var(--hty-text-color)" itemprop="url" rel="index"><span itemprop="text">acwing</span></a></span></span><span class="post-tag"><span class="post-meta-divider">-</span><a class="tag" href="/tags/%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84/" style="--text-color:var(--hty-text-color)"><span class="post-meta-item-icon"><svg class="icon" aria-hidden="true"><use xlink:href="#icon-price-tag-3-line"></use></svg></span><span class="tag-name">数据结构</span></a></span></div></div></header><section class="post-body" itemprop="articleBody"><div class="post-content markdown-body" style="--smc-primary:black;"><h3 id="第5讲-二叉排序树、表达式树"><a href="#第5讲-二叉排序树、表达式树" class="headerlink" title="第5讲 二叉排序树、表达式树"></a>第5讲 二叉排序树、表达式树</h3><h4 id="1-二叉排序树-BST"><a href="#1-二叉排序树-BST" class="headerlink" title="1.二叉排序树 BST"></a>1.二叉排序树 BST</h4><p>中序遍历有序的二叉树 为 二叉排序树</p>
<p><a href="https://www.acwing.com/problem/content/3789/" target="_blank" rel="noopener"><strong>AcWing 3786. 二叉排序树</strong></a></p>
<h4 id="2-平衡树——AVL"><a href="#2-平衡树——AVL" class="headerlink" title="2.平衡树——AVL"></a>2.平衡树——AVL</h4><p>(1) 定义：满足如下条件的树：<br>    a. 是二叉查找树<br>    b. 每个节点的左子树和右子树的高度差最多为1<br>(2) 平衡因子：一个结点的左子树的高度减去右子树的高度，可取-1、0、1三种值<br>(3) 平衡操作</p>
<blockquote>
<p>左旋 右旋 不改变中序遍历</p>
<p><strong>中序遍历不变</strong></p>
</blockquote>
<p><img src="https://gitee.com/li_zikang/lzk-image/raw/master/img/image-20210727163412049.png" alt="image-20210727163412049" loading="lazy"></p>
<h4 id="3-表达式树"><a href="#3-表达式树" class="headerlink" title="3.表达式树"></a>3.表达式树</h4><blockquote>
<p>考的少</p>
</blockquote>
<p><img src="https://gitee.com/li_zikang/lzk-image/raw/master/img/image-20210727164520285.png" alt="image-20210727164520285" loading="lazy"></p>
<h4 id="4-考题：2011-7、2012-4、2013-3-PDF中的分析有误，以上课讲解为准-、2013-6、2015-4、2018-6、2019-4、2019-6、2020-5、2017-41"><a href="#4-考题：2011-7、2012-4、2013-3-PDF中的分析有误，以上课讲解为准-、2013-6、2015-4、2018-6、2019-4、2019-6、2020-5、2017-41" class="headerlink" title="4.考题：2011-7、2012-4、2013-3(PDF中的分析有误，以上课讲解为准)、2013-6、2015-4、2018-6、2019-4、2019-6、2020-5、2017-41"></a>4.考题：2011-7、2012-4、2013-3(PDF中的分析有误，以上课讲解为准)、2013-6、2015-4、2018-6、2019-4、2019-6、2020-5、2017-41</h4><p><strong>2011-7</strong></p>
<p><img src="https://gitee.com/li_zikang/lzk-image/raw/master/img/image-20210727164907499.png" alt="image-20210727164907499" loading="lazy"></p>
<blockquote>
<p>A</p>
<p><strong>判断中序遍历是否有序</strong></p>
<p><img src="https://gitee.com/li_zikang/lzk-image/raw/master/img/image-20210727165303010.png" alt="image-20210727165303010" loading="lazy"></p>
<p><img src="https://gitee.com/li_zikang/lzk-image/raw/master/img/image-20210727165130394.png" alt="image-20210727165130394" loading="lazy"></p>
</blockquote>
<p><strong>2012-4</strong></p>
<p><img src="https://gitee.com/li_zikang/lzk-image/raw/master/img/image-20210727165415989.png" alt="image-20210727165415989" loading="lazy"></p>
<blockquote>
<p>B</p>
<p><img src="https://gitee.com/li_zikang/lzk-image/raw/master/img/image-20210727165720445.png" alt="image-20210727165720445" loading="lazy"></p>
</blockquote>
<p><strong>2013-3</strong></p>
<p><img src="https://gitee.com/li_zikang/lzk-image/raw/master/img/image-20210727163607794.png" alt="image-20210727163607794" loading="lazy"></p>
<blockquote>
<p>D    左旋 右旋 不改变中序遍历</p>
<p><img src="https://gitee.com/li_zikang/lzk-image/raw/master/img/image-20210727164002437.png" alt="image-20210727164002437" loading="lazy"></p>
</blockquote>
<p><strong>2013-6</strong></p>
<p><img src="https://gitee.com/li_zikang/lzk-image/raw/master/img/image-20210727165759234.png" alt="image-20210727165759234" loading="lazy"></p>
<blockquote>
<p>C</p>
<p>删除一个节点后又添加这个节点，树会不会不同</p>
<p>为叶节点： 相同</p>
<p>不为叶节点：</p>
<p><img src="https://gitee.com/li_zikang/lzk-image/raw/master/img/image-20210727170031782.png" alt="image-20210727170031782" loading="lazy"></p>
</blockquote>
<p><strong>2015-4</strong></p>
<p><img src="https://gitee.com/li_zikang/lzk-image/raw/master/img/image-20210727170408836.png" alt="image-20210727170408836" loading="lazy"></p>
<blockquote>
<p>D</p>
<p>AVL 树 中序遍历为降序</p>
<p>左子树一定大于此元素</p>
<p>最大元素一定无左子树</p>
</blockquote>
<p><strong>2018-6</strong></p>
<p><img src="https://gitee.com/li_zikang/lzk-image/raw/master/img/image-20210727170610360.png" alt="image-20210727170610360" loading="lazy"></p>
<p><img src="https://gitee.com/li_zikang/lzk-image/raw/master/img/image-20210727170617758.png" alt="image-20210727170617758" loading="lazy"></p>
<blockquote>
<p>C </p>
</blockquote>
<p><strong>2019-4</strong></p>
<p><img src="https://gitee.com/li_zikang/lzk-image/raw/master/img/image-20210727170743812.png" alt="image-20210727170743812" loading="lazy"></p>
<blockquote>
<p>A</p>
<p><img src="https://gitee.com/li_zikang/lzk-image/raw/master/img/image-20210727170958283.png" alt="image-20210727170958283" loading="lazy"></p>
<p><img src="https://gitee.com/li_zikang/lzk-image/raw/master/img/image-20210727171144035.png" alt="image-20210727171144035" loading="lazy"></p>
</blockquote>
<p><strong>2019-6</strong></p>
<p><img src="https://gitee.com/li_zikang/lzk-image/raw/master/img/image-20210727164634164.png" alt="image-20210727164634164" loading="lazy"></p>
<blockquote>
<p>A</p>
<p>表达式树的化简 </p>
<p><img src="https://gitee.com/li_zikang/lzk-image/raw/master/img/image-20210727164716317.png" alt="image-20210727164716317" loading="lazy"></p>
<p>到</p>
<p><img src="https://gitee.com/li_zikang/lzk-image/raw/master/img/image-20210727164737507.png" alt="image-20210727164737507" loading="lazy"></p>
</blockquote>
<p><strong>2020-5</strong></p>
<p><img src="https://gitee.com/li_zikang/lzk-image/raw/master/img/image-20210727171222387.png" alt="image-20210727171222387" loading="lazy"></p>
<p><strong>2017-41</strong></p>
<p><img src="https://gitee.com/li_zikang/lzk-image/raw/master/img/image-20210727171446721.png" alt="image-20210727171446721" loading="lazy"></p>
<blockquote>
<p><a href="https://www.acwing.com/problem/content/3768/" target="_blank" rel="noopener"><strong>AcWing 3765. 表达式树</strong></a>  </p>
</blockquote>
<p>C++：  $O(n^2)$</p>
<figure class="highlight c++"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br></pre></td><td class="code"><pre><code class="hljs c++"><span class="hljs-comment">/**</span><br><span class="hljs-comment"> * Definition for a binary tree node.</span><br><span class="hljs-comment"> * struct TreeNode &#123;</span><br><span class="hljs-comment"> *     string val；</span><br><span class="hljs-comment"> *     TreeNode *left;</span><br><span class="hljs-comment"> *     TreeNode *right;</span><br><span class="hljs-comment"> * &#125;;</span><br><span class="hljs-comment"> */</span><br><span class="hljs-class"><span class="hljs-keyword">class</span> <span class="hljs-title">Solution</span> &#123;</span><br><span class="hljs-keyword">public</span>:<br>    <br>    <span class="hljs-function"><span class="hljs-built_in">string</span> <span class="hljs-title">dfs</span><span class="hljs-params">(TreeNode* root)</span></span>&#123;<br>        <br>        <br>        <span class="hljs-keyword">if</span>(!root) <span class="hljs-keyword">return</span> <span class="hljs-string">""</span> ;<span class="hljs-comment">// 节点为空</span><br>        <span class="hljs-keyword">if</span>(!root-&gt;left&amp;&amp;!root-&gt;right) <span class="hljs-keyword">return</span> root-&gt;val;  <span class="hljs-comment">// 叶节点 </span><br>        <span class="hljs-keyword">else</span> &#123;<br>            <br>            <span class="hljs-keyword">return</span> <span class="hljs-string">"("</span> + dfs(root-&gt;left) + root-&gt;val + dfs(root-&gt;right) + <span class="hljs-string">")"</span>;<br>            <br>        &#125;<br>        <br>        <br>    &#125;<br>    <br>    <span class="hljs-function"><span class="hljs-built_in">string</span> <span class="hljs-title">expressionTree</span><span class="hljs-params">(TreeNode* root)</span> </span>&#123;<br>        <br>        <span class="hljs-comment">// 最外层不要括号</span><br>        <br>        <span class="hljs-keyword">return</span> dfs(root-&gt;left ) + root-&gt;val + dfs(root-&gt;right);<br>        <br>    &#125;<br>&#125;;<br></code></pre></td></tr></table></figure>

<p><img src="https://gitee.com/li_zikang/lzk-image/raw/master/img/image-20210727172012391.png" alt="image-20210727172012391" loading="lazy"></p>
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